"True possession is when a creature displaces or overrides
the target’s consciousness with its own, establishing
direct control over the target’s body. "

"Incorporeal Possession: Ghosts, shadow demons,
and similar creatures do not possess physical bodies—
they are simply disembodied souls. They use the rules
below to govern the interaction between their mental
characteristics and their hosts’ physical forms. Such a
creature merges with the host’s body and is unharmed
when ejected. Creatures that use magic jar can often use
the new possession spell instead"

Does any Incorporeal creature get to do this?
How does it work?
Just invade their body?

Not being a physical creature anymore, do they need to eat? If so, how?

Incorporeal creatures are almost always undead. Undead do not breathe, eat, or sleep (unless otherwise noted in their entry), incorporeal or not. Shadow demons aren't undead, but they are outsiders and not subtyped native, so they don't eat or sleep either.

I forget whether there are any incorporeals that neither undead nor outsiders.

1. No. The quoted paragraph merely explains the mechanism by which incorporeal creatures that happen to have the ability to possess others works. Many incorporeal creatures have no ability to possess anybody, including any bog-standard adventure using a belt of shadow form.

2. When it works at all, it works as the chapter you've quoted describes--unless the creature in question has its own special rules for possession, of course.

3. See 2.

4. If the possessed host body required food before possession, it likely still requires food after possession. If the possessed body was a corpse, that may not apply.

I forget whether there are any incorporeals that neither undead nor outsiders.

The only one I could find that wasn't a 3rd party creature was the Etheric Dragon luckly for it, all of it's natural attacks have the ghost touch property meaning it can still eat the same sort of things normal dragons eat.

So, it does seem to indicate that incorporeal or not if you are a creature type that needs to eat and/or breathe then you still need to find a way to do these things.

The Incorporeal Possession states that ghosts can possess someone.
None of the ghost creature entries state this ability.

Source PZO9446H:
Possession occurs when one creature, the possessor, forcibly displaces or overrides the consciousness of another creature, the host, with its own. True possession occurs only when a possessor gains direct control over another creature’s body. Effects that allow a creature to merely commandeer another being’s mind (such as dominate monster) aren’t true possession effects. Two general types of true possession exist: spell possession and incorporeal possession. Magic jar and possession are both examples of spell possession, magic that allows the caster to detach her soul from her body and invade her host’s body with it. In such cases, the caster’s body is often left behind—a glaring vulnerability for the possessor’s enemies to exploit. Incorporeal possession occurs when a creature without a physical body (such as a ghost or shadow demon) merges with the body of its host. In this event, the possessor leaves no trace of itself behind for foes to target.

Does an incorporeal creature (a ghost and similar), being a disembodied soul, get to take over a host simply by merging into it?

No, you need to have some sort of possession ability to possess a creature. The rule merely states what happens when an incorporeal creature does so (they don't leave a body behind). Incorporeal creatures cannot merge with other creatures merely by entering their square, you merge when you use possession.

Never said entering their square, merging with the host is what I said, as the quoted text states a ghost does. how does a ghost do it then? They don't have any ability listed that allows it, but the incorporeal possession says a ghost can.

Never said entering their square, merging with the host is what I said, as the quoted text states a ghost does. how does a ghost do it then? They don't have any ability listed that allows it, but the incorporeal possession says a ghost can.

They'd need to either have enough class levels of a spell casting class to have possession, or have an ability that allows them to do so, otherwise they can't possess (and thus merge).

It's likely that since ghosts are known for possession in fiction, the writer included them as an example on accident, thinking they had the ability to do so in pathfinder.

ghost is a template that gets applied to a creature. Which grants the creature special attacks based on the final CR of the ghost.

Ghost wrote:

Special Attacks: A ghost retains all the special attacks of the base creature, but any relying on physical contact do not function. In addition, a ghost gains one ghost special attack from the list below for every 3 points of CR (minimum 1—the first ability chosen must always be corrupting touch).

The ability that allows a ghost to use magic jar on creatures is malevolence.

Malevolence (Su) wrote:

The ghost’s jealousy of the living is particularly potent. Once per round, the ghost can merge its body with a creature on the Material Plane. This ability is similar to a magic jar spell (caster level 10th or the ghost’s Hit Dice, whichever is higher), except that it does not require a receptacle. To use this ability, the ghost must be adjacent to the target. The target can resist the attack with a successful Will save. A creature that successfully saves is immune to that same ghost’s malevolence for 24 hours.

Occult adventures provides alternate rules that a DM may use if they wish substituting magic jar for possession. Because the rules are optional, none of the monster entries have been updated to reflect this change since its not the default. Additionally, not all ghosts get the malevolence ability. A Ghost of CR 5 or less is incapable of picking it up since they only get one ability from the list and must choose corrupting touch. Conversely, since there are 6 abilities in said list you could have a CR 17 Ghost that also lacks it. It all just depends on how the DM has built the creature.

Never said entering their square, merging with the host is what I said, as the quoted text states a ghost does. how does a ghost do it then? They don't have any ability listed that allows it, but the incorporeal possession says a ghost can.

They'd need to either have enough class levels of a spell casting class to have possession, or have an ability that allows them to do so, otherwise they can't possess (and thus merge).

It's likely that since ghosts are known for possession in fiction, the writer included them as an example on accident, thinking they had the ability to do so in pathfinder.

The quoted text is from the Haunted Heroes Handbook, and the original ghost stat block is from Bestiary, predating it by more than a decade. I'd just assumed that the Haunted Heroes Handbook itself had more context for how ghosts were different in occult and/or horror games. AoN doesn't have all of the text of every book, after all.